Determination of the Optimally Efficient Sets in Special Classes of Graphs

Mihai Tălmaciu“Vasile Alecsandri” University
Bacău

Abstract:

A dominating set is said to be an efficient dominating set if, for every vertex v ∈ V, |N[v] ∩ S| = 1 [1]. A set S is called open irredundant if, for every vertex u ∈ S, there exists a vertex v ∈ V ? S for which N(v) ∩ S = {u}, in which case we say that u efficiently dominates v. There exists a polynomial time algorithm for finding an optimally efficient set in an arbitrary graph. We determine directly the optimally efficient sets in confidentially connected graphs and unbreakable graphs. Also, we determine directly the open irredundant set, closed neighborhood packing set, the influence of a set in confidentially connected graphs and unbreakable graphs.

This article is motivated because there is only a polynomial time algorithm for finding an optimally efficient set in an arbitrary graph, and we determine it directly for special classes of graphs. Also, in [13] domination and irredundance parameters for some graphs have been extensively studied. For a survey see [9].

The efficiency of a set S ⊆ V in a graph G=(V,E), is defined as ε(S) = |{v ∈ V −S: |N(v)∩ S| = 1}|. The efficiency of a graph, denoted ε(G), is defined to equal the maximum efficiency of a set S ⊆ V in G [3]. A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge. That is, a set D is a dominating set if |N[v] ∩ D| ≥ 1 for all vertices v in V (G). The domination number γ(G) is the number of vertices in the smallest dominating set of G. A subset S V (G) is called a k -packing, if for any two distinct vertices u, v in S , we have d(u, v) > k. A set S is a closed neighborhood packing if for each u, vS, uv we have N[u] ∩ N[v] = . That is, a set S is a closed neighborhood packing if |N[v] ∩ S| ≤ 1 for all vertices vV (G). The packing numberρ(G) is the size of the largest closed neighborhood packing. For all graphs G, 1 ≤ ρ(G) ≤ n . The only graphs with ρ(G) = n are graphs with no edges.

A dominating set S is called a perfect dominating set if every vertex v ∈ V − S is adjacent to exactly one vertex in S [4]. A dominating set is said to be an efficient dominating set if for every vertex v ∈ V, |N[v] ∩ S| = 1 [1]. The efficient domination number of a graph, denoted F(G), is the maximum number of vertices that can be dominated by a set S that dominates each vertex at the most once. A graph G of order n = |V (G)| has an efficient dominating set if and only if F(G) = n.

A graph is efficient if and only if there exists an efficient dominating set. That is, a graph is efficient if and only if there exists a set S which is both dominating and a closed neighborhood packing.

If a graph G is efficient, then ρ(G) = γ(G) [12] .

The influence of S is defined in [8] to be
I(S) = ,

where deg(v) = |N(v)|, the cardinality of the open neighborhood of v.

Thus, the efficient domination numberof a graph G is F(G)=max{I(S): S is a packing}=max{ and u, v S implies d(u, v) 3}. An F(G) -set S is a set that is both a packing and I(S) = F(G).

A set S is called open irredundant if for every vertex u ∈ S there exists a vertex v ∈ V − S for which N(v) ∩ S = {u}, in which case we say that u efficiently dominates v [7].

The upper open irredundance number, denoted OIR(G), equals the maximum number of vertices in an open irredundant set. Thus, OIR(G) equals the maximum number of vertices that can simultaneously and successfully broadcast a message in an Ethernet graph.

By contrast, the efficiency of a graph G equals the maximum number of vertices that can simultaneously receive a broadcast message in an Ethernet graph.

In a network, the communication is said to be “confidential” if a message can be passed between any two vertices, without being intercepted by a third vertex. In the language of graph theory, this property stands for confidential connectivity.

There exists a polynomial time algorithm for finding an optimally efficient set in an arbitrary graph [10].